Why is vector $[A,B]$ perpendicular to line $Ax + By + C = 0$ and vector $[-B, A]$ is parallel to it? I've seen some proofs, but I don't know what is the intuition behind this. After thinking about this form of representation of a line I can't see why this equation represents a line and why does those vectors are perpendicular/parallel. Can someone give me some intuition how to think about it?
[Math] Why is vector $[A,B]$ perpendicular to line $Ax + By + C = 0$
vectors
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Best Answer
Write it as,
$$A(x-h)+B(y-k)=0$$
Let $\vec v=\langle A,B \rangle$ and $\vec w= \langle h,k \rangle$ and $\vec x=\langle x,y \rangle$.
Then,
$$\vec v \cdot (\vec x-\vec w)=0$$
The tail of position vector $\vec w$ lies on the line $Ax+By=C$. If you draw $\vec x-\vec w$ you will see why this shows $\vec v$ is orthogonal to the line.
Now as two why $\langle -B, A \rangle$ is a direction vector. You can deduce this from the fact that $\langle -B,A \rangle \cdot \langle A,B \rangle=0$.